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Math Help - pointwise limit?

  1. #1
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    Exclamation pointwise limit?

    Let Sn(x)= nx/(nx+1)

    Compute the pointwise limit of Sn on [0,1].
    Is the convergence uniform?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by cowgirl123 View Post
    Let Sn(x)= nx/(nx+1)

    Compute the pointwise limit of Sn on [0,1].
    Is the convergence uniform?
    S_n \to S point-wise on [0,1] where S = \left \{ \begin{array}{lr} 0 & \mbox{ if } x = 0 \\ & \\ 1 & \mbox{ if } x \in (0,1] \end{array} \right.

    all we did was to take the limit as n \to \infty



    Is it uniform? there are several ways to attack this.

    what do we know about the limit of a uniformly continuous function?

    we could also use the \epsilon - \delta definition for uniform convergence

    we could also check to see whether or not \lim_{n \to \infty} [ \sup \{ |S(x) - S_n(x)|~:~x \in [0,1] \}] = 0 in which case it is uniformly convergent if it is 0
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